Pyramidal number

Pyramid number is a spatial type of figure numbers representing a pyramid with a polygonal base and a given number of triangular sides. Already ancient mathematicians investigated tetrahedral and square pyramidal numbers, for which at the base lie a regular triangle and a square, respectively. It is easy to determine the numbers associated with the pyramids , the base of which is any other polygon, for example:

Geometric representation of a square pyramidal number:

\Pi _{4}^{(4)}=1+4+9+16=30

{\ displaystyle \ Pi _ {4} ^ {(4)} = 1 + 4 + 9 + 16 = 30}

{\ displaystyle \ Pi _ {4} ^ {(4)} = 1 + 4 + 9 + 16 = 30}

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Content

Definition

Pyramidal numbers are defined as follows.

$n$ ${\ displaystyle n}$ in order $k$ ${\ displaystyle k}$ angular pyramid number $\Pi _{n}^{(k)}$ ${\ displaystyle \ Pi _ {n} ^ {(k)}}$ there is a sum of first $n$ ${\ displaystyle n}$ flat shaped numbers $P_{n}^{(k)}$ ${\ displaystyle P_ {n} ^ {(k)}}$ with the same number of angles $k$ ${\ displaystyle k}$ :

\Pi _{n}^{(k)}=P_{1}^{(k)}+P_{2}^{(k)}+P_{3}^{(k)}+\dots +P_{n}^{(k)}

{\ displaystyle \ Pi _ {n} ^ {(k)} = P_ {1} ^ {(k)} + P_ {2} ^ {(k)} + P_ {3} ^ {(k)} + \ dots + p_ {n} ^ {(k)}}

Geometrically pyramidal number $\Pi _{n}^{(k)}$ ${\ displaystyle \ Pi _ {n} ^ {(k)}}$ can be represented as a pyramid of $n$ ${\ displaystyle n}$ layers (see drawing), each of which contains from 1 (top layer) to $P_{n}^{(k)}$ ${\ displaystyle P_ {n} ^ {(k)}}$ (bottom) balls.

By induction, it is not difficult to prove the general formula for a pyramid number, also known to Archimedes ^[1] :

\Pi _{n}^{(k)}={\frac {n(n+1)((k-2)n-k+5)}{6}}

{\ displaystyle \ Pi _ {n} ^ {(k)} = {\ frac {n (n + 1) ((k-2) n-k + 5)} {6}}

(OPF)

The right side of this formula can also be expressed in terms of flat polygonal numbers:

\Pi _{n}^{(k)}={\frac {(k-2)n-k+5}{3}}P_{n}^{(3)}={\frac {n+1}{6}}(2P_{n}^{(k)}+n)

{\ displaystyle \ Pi _ {n} ^ {(k)} = {\ frac {(k-2) n-k + 5} {3}} P_ {n} ^ {(3)} = {\ frac { n + 1} {6}} (2P_ {n} ^ {(k)} + n)}

Notes

↑ Desa E., Desa M., 2016 , p. 70-71.

Literature

Vilenkin N. Ya., Shibasov L. P. Shibasova 3. F. Behind the pages of a textbook of mathematics: Arithmetic. Algebra. Geometry. - M .: Enlightenment, 1996. - p. 30. - 320 p. - ISBN 5-09-006575-6 .
Glazer GI. The history of mathematics in school . - M .: The Enlightenment, 1964. - 376 p.
Desa E., Desa M. Figured numbers. - M .: MTSNMO, 2016. - 349 p. - ISBN 978-5-4439-2400-7 .

Links

Figured numbers
Figurate Numbers on MathWorld (English)
Centered Polygonal Number at MathWorld (English)

[_e7e8729a69c8fae0-1] Desa E., Desa M., 2016 , p. 70-71.